Mean-field models for disordered crystals

In this article, we set up a functional setting for mean-field electronic structure models of Hartree-Fock or Kohn-Sham types for disordered quantum systems. In the first part, we establish important properties of stochastic fermionic one-body density matrices, assuming that they are stationary under the ergodic action of a translation group. In particular, we prove Hoffmann-Ostenhof and Lieb-Thirring inequalities for ergodic density matrices, and deduce some weak compactness properties of the set of such matrices. We also discuss the representability problem for the associated one-particle densities. In the second part, we investigate the problem of solving Poisson's equation for a given stationary charge distribution, using the Yukawa potential to appropriately define the Coulomb self-interaction in the limit when the Yukawa parameter goes to zero. Finally, in the last part of the article, we use these tools to study a specific mean-field model (reduced Hartree-Fock, rHF) for a disordered crystal where the nuclei are classical particles whose positions and charges are random. We prove the existence of a minimizer of the energy per unit volume and the uniqueness of the ground state density. For (short-range) Yukawa interactions, we prove in addition that the rHF ground state density matrix satisfies a self-consistent equation, and that our model is the thermodynamic limit of the supercell model.